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Properties

Label 230.6.b.a
Level $230$
Weight $6$
Character orbit 230.b
Analytic conductor $36.888$
Analytic rank $0$
Dimension $26$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [230,6,Mod(139,230)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("230.139"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(230, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 230 = 2 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 230.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [26] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(36.8882785570\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 26 q - 416 q^{4} - 30 q^{5} - 72 q^{6} - 1400 q^{9} + 80 q^{10} - 1314 q^{11} + 808 q^{14} + 1280 q^{15} + 6656 q^{16} + 6630 q^{19} + 480 q^{20} - 10060 q^{21} + 1152 q^{24} - 10470 q^{25} - 376 q^{26}+ \cdots + 523850 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1 − 4.00000i − 27.5301i −16.0000 28.4300 − 48.1325i −110.120 − 119.184i 64.0000i −514.906 −192.530 − 113.720i
139.2 − 4.00000i − 26.8618i −16.0000 41.5913 + 37.3519i −107.447 117.217i 64.0000i −478.554 149.408 − 166.365i
139.3 − 4.00000i − 17.9630i −16.0000 −33.0651 + 45.0744i −71.8519 13.4457i 64.0000i −79.6687 180.297 + 132.260i
139.4 − 4.00000i − 13.9839i −16.0000 −7.76744 − 55.3594i −55.9357 − 114.735i 64.0000i 47.4501 −221.438 + 31.0698i
139.5 − 4.00000i − 11.2210i −16.0000 −42.1119 + 36.7639i −44.8839 166.669i 64.0000i 117.090 147.056 + 168.448i
139.6 − 4.00000i − 5.96211i −16.0000 28.0011 + 48.3833i −23.8485 − 183.563i 64.0000i 207.453 193.533 − 112.004i
139.7 − 4.00000i 1.04517i −16.0000 −54.3251 − 13.1829i 4.18067 − 135.681i 64.0000i 241.908 −52.7315 + 217.300i
139.8 − 4.00000i 5.15388i −16.0000 33.2902 − 44.9084i 20.6155 47.7267i 64.0000i 216.438 −179.633 − 133.161i
139.9 − 4.00000i 11.0852i −16.0000 −42.4833 − 36.3342i 44.3407 240.887i 64.0000i 120.119 −145.337 + 169.933i
139.10 − 4.00000i 14.1507i −16.0000 55.0409 + 9.77250i 56.6028 1.53351i 64.0000i 42.7574 39.0900 − 220.164i
139.11 − 4.00000i 18.6563i −16.0000 −47.1490 + 30.0329i 74.6251 68.7947i 64.0000i −105.057 120.131 + 188.596i
139.12 − 4.00000i 19.5707i −16.0000 11.6274 + 54.6791i 78.2830 − 53.5462i 64.0000i −140.014 218.716 − 46.5095i
139.13 − 4.00000i 24.8599i −16.0000 13.9210 − 54.1406i 99.4396 51.4359i 64.0000i −375.015 −216.562 − 55.6841i
139.14 4.00000i − 24.8599i −16.0000 13.9210 + 54.1406i 99.4396 − 51.4359i − 64.0000i −375.015 −216.562 + 55.6841i
139.15 4.00000i − 19.5707i −16.0000 11.6274 − 54.6791i 78.2830 53.5462i − 64.0000i −140.014 218.716 + 46.5095i
139.16 4.00000i − 18.6563i −16.0000 −47.1490 − 30.0329i 74.6251 − 68.7947i − 64.0000i −105.057 120.131 − 188.596i
139.17 4.00000i − 14.1507i −16.0000 55.0409 − 9.77250i 56.6028 − 1.53351i − 64.0000i 42.7574 39.0900 + 220.164i
139.18 4.00000i − 11.0852i −16.0000 −42.4833 + 36.3342i 44.3407 − 240.887i − 64.0000i 120.119 −145.337 − 169.933i
139.19 4.00000i − 5.15388i −16.0000 33.2902 + 44.9084i 20.6155 − 47.7267i − 64.0000i 216.438 −179.633 + 133.161i
139.20 4.00000i − 1.04517i −16.0000 −54.3251 + 13.1829i 4.18067 135.681i − 64.0000i 241.908 −52.7315 − 217.300i
See all 26 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 139.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.6.b.a ✓ 26
5.b even 2 1 inner 230.6.b.a ✓ 26
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.6.b.a ✓ 26 1.a even 1 1 trivial
230.6.b.a ✓ 26 5.b even 2 1 inner